Diane- that's simple - a bedsheet and some pushpins. Or better, if you have some spare light stands, or can get some, put up two light stands and stretch a piece of rope between them, then hang the sheet over the rope. Easy-peasy.
Yep--pushpins work well with a bedsheet, and if you invest in a few of those spring clamps that look like big metal clothespins with orange handles, you can hang a blanket from a doorframe, door, or whatever. The only problem with the light-stand-and-rope approach is that unless you use something rigid across the top (plastic sprinkler pipe, for example) the weight of the backdrop will try to tip the stands inward.
You are probably not talking about a large field of view (a couple or three feet?) so you can get away with setting up the camera first, looking to see what is in the field, and then just draping that area.
My current backdrop support is cup hooks into the ceiling joists, hanging a piece of electrical conduit (EMT, the thin-walled stuff) from loops of twine, and using the aforementioned clamps to attach a Velux blanket from Wal-mart. I originally thought that I would upgrade to plant-hanger hooks, but the setup works so well that I never bothered.
I've also used scraps of dollar-a-yard fabric from Wal-Mart to make "seamless" backdrops by taping it to the wall (careful of the paint...) and letting it drape out over a tabletop.
The possibilities are endless...